On the Statistical Distribution of Stationary
Segment Lengths of Road Vehicles Vibrations
Vincent Rouillard
Abstract— This paper presents an important outcome of a research programme which focuses on the development of a method for synthesizing, under controlled conditions in the laboratory, the non-stationary random vibrations generated by road transport vehicles. It addresses an important limitation of current methods used for synthesising random vehicle vibrations which assume that vibrations produced by wheeled vehicles can be approximated by a zero-mean, normally-distributed (Gaussian) random process and, therefore, fails to accurately reproduce the fluctuations in vibration levels that occur naturally during road transportation realizations [1]. The paper builds upon the observation that non-stationary random vehicle vibrations are composed of a sequence of zero-mean random Gaussian processes of varying standard deviations [2]. It discusses the important parameters that need to be addressed when dealing with the synthesis of random sequences. The paper presents the development of a change-point detection algorithm that was used to determine the length of stationary segments within a large number of typical non-stationary random vibration records. These include measured vibration records as well as numerically-generated records based on measured pavement profiles. The algorithm, based on the cum-sum / bootstrapping techniques, was developed and applied to the instantaneous magnitude of sample vibration records. The statistical distribution of segment lengths for each vibration record was computed with the aim of identifying similarities and trends for the development of an overall statistical model for segment lengths to be used for synthesis purposes. One outcome of note was that the shape of the segment length distributions computed from a wide range of vibration records are generally comparable and exhibit an asymptotic-like decrease in probability of occurrence as the segment length increases. This behaviour was found to be adequately modelled with a hyperbolic trigonometric function which was found to be useful for characterising the general statistical behaviour of segment length for non-stationary random vibrations produced by road vehicles. Finally, the significance and relevance of this outcome with respect to the synthesis of non-stationary vibrations for package evaluation and validation purposes is highlighted.
Index Terms—Non-stationary vibrations, random vibrations, vehicle vibrations, change-points, segment lengths.
Author is with Victoria University, Melbourne, PO Box 14428, MCMC 8001, Australia. (phone: +613 9919 4602; fax: +613 9919 4139; e-mail: Vincent.Rouillard@vu.edu.au).
I. INTRODUCTION
It is self-evident that vehicle vibrations are caused, in the main, by uneven pavement surfaces. These have been found to be random in nature which, in turn, causes the vehicle vibrations to be random. Variations in pavement roughness and fluctuations in the vehicle’s speed within a particular journey combine to produce variations in the overall energy levels of the vehicle vibrations. These variations make the process highly non-stationary [2, 3] and introduce a level of complexity that cannot be adequately dealt with using conventional methods that are used for Gaussian and stationary processes. The most commonly-used technique for laboratory simulation of transport vibrations has been in place for some years. The method assumes that the vibrations produced by wheeled vehicles can be approximated by a zero-mean, normally-distributed (Gaussian) random process. In addition, the overall root-mean-square (rms) level of the process is often assumed to be constant thus implying stationarity. It has been shown that vibration synthesis at a constant rms level fail to accurately reproduce the fluctuations in vibration levels that occur naturally during road transportation realizations [1]. Most laboratories make use of a random vibration controller (RVC) which uses a feedback loop to synthesize and control random signals according to a PSD function. When non-stationarity is taken into account, vibrations are synthesized at various rms levels for pre-determined durations. However, the length and sequence of each constant rms synthesized segment is not known. This is the outstanding issue addressed herein.
Gaussian segment will be synthesized.
II. CHANGE POINT DETECTION
Change-point detection is a process that emerged out of a need to identify real changes in random processes such as economic indicators and the monitoring and control of quality in manufacturing processes. Techniques range from cumulative sum (cum-sum) schemes first proposed by Page [4, 5, 6] and further developed by Hinkley [7] and Pettitt [8] to Singular-Spectrum Analysis [9]. The cumulative sum techniques are primarily geared toward detecting significant deviations in the mean of random processes. The procedure is greatly enhanced by including a bootstrapping algorithm which is used to provide an estimate in the significance of the change-point. Bootstrapping effectively rearranges the sequence or order of the sample in a random fashion a number of times while re-evaluating the sample (in this case using the cum-sum algorithm). This gives a basis for determining whether the change is truly significant or merely apparent. In this study, a change-point detection algorithm, based on the cum-sum / bootstrapping techniques was developed and applied to the instantaneous magnitude of the sample vibration records. The argument for using the instantaneous magnitude are based on evidence which shows that level type non-stationarities in random signals are well manifested through changes in the instantaneous magnitude and are not reliant on subjective parameters such as the window width required to compute the moving RMS [10].
The instantaneous magnitude accounts for the deficiencies with the moving RMS method in detecting short duration fluctuations in the magnitude of vehicle vibrations which are typically induced when the vehicle encounters sudden changes in the roughness of pavements and when severe and localised pavement surface defects are present [11]. The method presented here makes use of the Hilbert transform to compute the instantaneous magnitude of the record. In its original form, the Hilbert Transform is used to produce the imaginary component, ã(t), of a measured, real signal, a(t), thus enabling the creation of an analytical signal â(t) [12]:
( )
( )
( )
ˆa t =a t +i a t
(1)where the Hilbert transform is defined as:
( )
{ }
( )
( )
1 1
a t a t a( ) d
t
1 1
a t t
τ
τ
π
τ
π
∞−∞
⎛ ⎞
= = ⎜ ⎟
−
⎝ ⎠
⎛ ⎞
= ∗ ⎜ ⎟
⎝ ⎠
∫
H(2)
Executing the convolution in equation (2), it can be shown that the Hilbert transform is equivalent to a 90o phase shift of
the real signal. This is significant in that it allows the computation of the Hilbert transform to be performed in the frequency domain via the Fourier transform as follows:
{
}
[
]
a( t )
A
( f )
A( f )
j sgn( f )
=
⊥
=
⋅ −
(3)
where A(f) is the spectrum of a(t) This operation where the positive and negative frequency components of the signal are given a phase shift of –90o (multiplied by –j) and +90o (multiplied by j) respectively. Once the analytical signal is created, the instantaneous magnitude of the vibration,
M[a(t)], signal is easily computed:
( )
( )
2( )
2M a t⎡⎣ ⎤ =⎦ a t +a t
(4) The instantaneous magnitude of the signal is manifested as the envelope of the signal as illustrated in Fig. 1 which shows a typical vibration record together with itsinstantaneous magnitude computed from the Hilbert Transform. It demonstrates how the instantaneous magnitude is useful to describe short-term and rapid variations in the amplitude of the signal. The main benefit is that there is no averaging process employed and the true magnitude-based non-stationary character of the signal is revealed.
The change-point detection algorithm was developed specifically to deal with the amplitude type
non-stationarities that are prevalent in road vehicle vibrations. In order to test and validate the algorithm, a number of vehicle vibration samples were collected from a variety of vehicle types, routes and payload conditions as shown in Table 1. In addition, the vertical acceleration responses of various linear quarter-car numerical models, made available in the literature, were computed for a range of pavement profiles (Table 2). These vibration records were generated to supplement and complement the
Distance [m]
Acce
ler
at
io
n [m
/s
2]
[image:3.612.56.578.51.211.2]Instantaneous magnitude
[image:3.612.54.301.281.517.2]Figure 1. Example of the manifestation of a vibration signal’s instantaneous magnitude (dark line).
Table I. Summary of measured vibration record parameters.
ID Vehicle type & load Route Type
MA Utility vehicle (1 Tonne capacity). Load: < 5% cap.
S'urban streets MB Prime mover + Semi trailer (Air ride
susp.). Load: 90% cap.
Country roads MC Transport van (700 kg capacity). Load:
60% cap.
Suburban streets MD Transport van (700 kg capacity). Load:
60% cap.
Suburban hwy. ME Transport van (700 kg capacity). Load:
60% cap.
Motorway
MF Prime mover + Semi trailer (leaf spring susp.). Load: < 5% cap.
Country roads MG Tipper truck (16 Tonnes capacity, Air ride
susp.). Load: 25% capacity.
Country roads MH Small flat bet truck (1 Tonne capacity, leaf
spring susp.). Load <5% cap.
Suburban streets MJ Flat bed truck (5 Tonnes capacity, leaf
spring susp.). Load >95% cap.
Country roads MK Sedan car. Load: 1 passenger Suburban
Table II. Summary of routes used for numerically-generated vibration records.
ID Route (Victoria, Australia)
SA Murray Valley Highway (Major county road) SB Bendigo – Maryborough road (Major county road) SC Princess Highway (Freeway)
SD Timboon Road, Victoria, Australia (Major county road) SE Road sequence: Timboon Road – Princess Hwy and
Murray Valley Hwy, Victoria, Aust.
The change-point algorithm developed for this study and associated statistical analysis is described hereunder.
à The instantaneous magnitude (envelope) of vibration is computed using the Hilbert Transform
à Compute the cumulative sum (cum-sum) of the instantaneous magnitude vector normalised with respect to the mean magnitude.
à Apply the bootstrap algorithm sequence whereby the entire instantaneous vibration vector is randomly re-samples (shuffled) a number of times and the cum-sum re-computed for each re-sampled vector
à The maximum and minimum envelopes from the bootstrap (shuffled) samples are computed
à The largest extremum of the original record is detected and identified as a change-point. Its value is compared with that of the bootstrap sample as illustrated in Fig. 2.
à The change point is identified as significant or valid if the ratio of the largest extremum to the bootstrap extremum exceeds a pre-determined value. In all cases studied, the ratio threshold of 5.5 was identified as adequate. The ratio threshold
effectively controls the sensitivity of the algorithm. Large ratio threshold values results in low
sensitivity whereby small changes in instantaneous magnitude will not be detected. Small ratio threshold values will enable the algorithm to detect small changes in instantaneous magnitude. Given the stochastic nature of the processes under consideration coupled with the relatively broad range of frequencies present, too small a ratio threshold is problematic in that the algorithm will identify a large number of fictitious change-points within the natural fluctuations in a stationary segment.
à If the change point is identified as real or
stationary segments within the process. Fig.3 shows an example of the outcome of the algorithm for a typical vibration record.
Original record
Change point Extremum
Bootstrap extremum
Bootstrap cum-sum maximum envelope
[image:4.612.79.530.114.416.2]Bootstrap cum-sum minimum envelope
Figure 2. Determination of a change-point using the cum-sum / bootstrap algorithms.
Figure 3. Identification of stationary segments (change-points shown in dark) using the cum-sum / bootstrap algorithm for a section from a typical vibration record.
The change-point detection algorithm, written in Matlab® (available from the author on request), was used to
determine the length of stationary segments within a large number of typical non-stationary random vibration records. These include measured vibration records as well as numerically-generated records based on quarter-car models
[image:4.612.71.552.437.625.2]vibration record and one typical numerically-generated record are shown in Fig. 4 and 5 respectively.
Figure 4. Statistical distribution of stationary segment lengths for a typical measured vibration record along with
the hyperbolic line (dark) of best fit.
Figure 5. Statistical distribution of stationary segment lengths for a typical numerically-generated vibration record
along with the hyperbolic line (dark) of best fit. The statistical distributions of the segment lengths for all nine measured vibrations records and all thirteen
numerically-generated records are shown separately in Fig.
6. It is interesting to note that the shape of the segment length distributions are generally comparable and exhibit an asymptotic – like decrease in probability of occurrence as the segment length increases. This behaviour was found to be competently modelled with a hyperbolic function in the form
( )
(
p s =C / sinh ks
)
(5) where C and k are empirical constants obtained by a non-linear least squares regression method. This model was applied to the measured and numerically-generated vibration data sets separately yielding lines of best fit shown in Fig. 7.Measured Data (C = 2.5, k = 0.19)
0 5 10 15 20 25 30 35 40
0 5 10 15 20
Segment length [s]
P(s) [%
]
DATAMA DATAMB DATAMC DATAMD DATAME DATAMF DATAMG DATAMH DATAMK Hyperbolic fit
Numerically generated Data (C = 5.0, k = 0.31)
0 5 10 15 20 25 30 35 40 45
0 5 10 15 20
Segment length [s]
P(
s) [
%
]
SA1100 SA2100 SA3100 SA4100 SB1120 SB2120 SB3120 SB4120 SC180 SC280 SC380 SC480 SE180 Hyperbolic fit
[image:5.612.327.558.211.620.2]Figure 6. Statistical distributions of stationary segment lengths for all 9 typical measured vibration records (top) and 13 typical numerically-generated vibration records (bottom)
[image:5.612.66.281.402.667.2]distinguish one from the other as shown in Fig. 7 which shows that a single probability density function in the form of Eqn. (4) with C ≈ 4 and k ≈ 1/4 is sufficient for describing the collective expected occurrence of stationary segment lengths for road vehicle vibrations.
Combined Data (C = 4.1, k = 0.27)
0 5 10 15 20 25 30 35 40 45
0 5 10 15 20
Segment length [s]
P(s)
[
%
]
0.01 0.1 1 10 100
0 10 20 30
Segment length [s]
40
P(
s)
[%]
DATAMA DATAMB DATAMC DATAMD DATAME DATAMF DATAMG DATAMH DATAMK
[image:6.612.55.297.121.541.2]SA1100 SA2100 SA3100 SA4100 SB1120 SB2120 SB3120 SB4120 SC180 SC280 SC380 SC480 SE180 Hyperbolic fit
Figure 7. Statistical distributions of stationary segment lengths for all sample vibration records long with the hyperbolic curve of best fit. (Left: Linear scales, Right:
semi-logarithmic scales for clarity).
III. CONCLUSIONS
The effectiveness of a novel approach to detect significant change-points within records of non-stationary random vibrations produced by wheeled vehicles has been
demonstrated. The paper has shown that the instantaneous magnitude, easily computed by means of the Hilbert
transform, can be used to apply the cumulative cum (cum sum) /bootstrapping algorithm to detect statistically significant change pints within the process. The main relevance of the results is the ability to determine the length of stationary segments within the process segment as well as their statistical distribution. The similarity of form between the segment length probability densities obtained from a wide variety of conditions (routes and vehicles as well as experimentally measured and numerically-generated data from measured pavement topographies) is unexpected. This demonstrates that, in terms of analysis methodology, there is no need to differentiate between measured and numerically generated vibration data. Finally, the paper proposed a simple hyperbolic distribution model that quite satisfactorily characterises the probability density of segment lengths. This outcome is relevant when attempting to synthesize realistic, non-stationary random vibrations in the laboratory. This technique, to be reported at a later date, makes use of the segment length distribution function to create a modulation function which is then used to alter the overall root-mean-square of a synthesized Gaussian signal to reproduce the amplitude fluctuations that are manifest in the process.
REFERENCES
[1] Rouillard, V. & Sek, M.A.. Simulation of Non-stationary vehicle Vibrations. Proceedings of the Institution of Mechanical Engineers, Vol 215, Part D, 1069-1075, 2001
[2] Rouillard, V. Gaussian Decomposition of Non-Stationary Random Vehicle Vibrations. Proceedings of the 2007 World Congress on Engineering, London, UK. (in press)
[3] Richards, D.P. A Review of Analysis and Assessment Methodologies for Road Transportation Vibration and Shock Data. Environmental Engineering, 23-26, Dec. 1990
[4] Page, E.S. A Test for Change in a Parameter Occurring at an Unknown Point. Biometrika,Vol.41, No.1,2, pp 100 – 115, 1954. [5] Page, E.S. A Test for Change in a Parameter Occurring at an
Unknown Point. Biometrika, Vol. 42, No. 3,4, pp 523 – 527, 1955 [6] Page, E.S. On Problems in which a Change in a Parameter Occurs at
an Unknown Point. Biometrika, Vol. 44, No. 1,2, pp 248 – 252, 1957. [7] Hinkley, D.V. Inference about the Change-Point from Cummulative
Sum Tests. Biometrika, Vol. 58, No. 3, pp 509 – 523, 1971. [8] Pettitt, A.N. A Simple Cumulative Sum Type Statistic for the
Change-Point Problem with Zero-One Observations. Biometrika, Vol. 67, No. 1, pp 74 – 84, 1980.
[9] Moskvina, V. Application of the Singular Spectrum Anslysis for Change-Point Detection in Time Series. PhD Dissertation, School of Mathematics, Cardiff University, UK, 2001.
[10] Rouillard, V., Sek, M.A. & Perry, T. (1996) The Analysis and Simulation of Road Profiles. ASCE Journal of Transportation Engineering, vol. 122, No. 3, pp 241 – 245, 1996.
[11] Rouillard V. Using predicted ride quality to characterise pavement roughness. International Journal of Vehicle Design V36 N2 - 3, pp 116 – 131, 2004.
